Defining equation

determining equation, characteristic equation

An equation associated with a regular singular point of an ordinary linear differential equation

(1)

Let

where the functions are holomorphic at the point and . The defining equation takes the form:

(2)

If the roots , , of equation (2) are such that all differences , where , are not integers, then equation (1) has a fundamental system of solutions of the form

(3)

where the functions are holomorphic at . Otherwise the coefficients can be polynomials in with coefficients holomorphic at .

The defining equation for a system of equations

(4)

corresponding to the regular singular point , takes the form

where is a matrix-function of order , holomorphic at and . If all differences , where , are not integers, where the are the eigen values of , then the system (4) has a fundamental system of solutions of the form (3), where are vector-functions holomorphic at ; otherwise, the vector-functions can be polynomials in with coefficients which are vector-functions holomorphic at .

In another sense, the term "determining equation" is used in research on transformation groups admitted by ordinary partial differential equations (see [3]).

References

[1]	E.A. Coddington, N. Levinson, "Theory of ordinary differential equations" , McGraw-Hill (1955)
[2]	E. Kamke, "Differentialgleichungen: Lösungen und Lösungsmethoden" , 1. Gewöhnliche Differentialgleichungen , Chelsea, reprint (1971)
[3]	L.V. [L.V. Ovsyannikov] Ovsiannikov, "Group analysis of differential equations" , Acad. Press (1982) (Translated from Russian)

Comments

A defining equation is more commonly called an indicial equation.